Cooling+ Heating Flows in Galaxy Clusters: Turbulent Heating, Spectral

DRAFTVERSION AUGUST 6, 2019 Preprint typeset using LATEX style emulateapj v. 12/16/11

COOLING+HEATING FLOWS IN GALAXY CLUSTERS: TURBULENT HEATING, SPECTRAL MODELLING, AND COOLING EFFICIENCY

, , , , MOHAMMAD H.ZHOOLIDEH HAGHIGHI1 2 3,NIAYESH AFSHORDI2 3 4, AND HABIB.G.KHOSROSHAHI1 1School of Astronomy, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, Iran 2Perimeter Institute for Theoretical Physics, 31 Carolines St. North, Waterloo, ON, N2L 2Y5, Canada 3Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran 4Department of Physics and Astronomy, University of Waterloo, 200 University Ave. West, Waterloo, ON, N2L 3G1, Canada

Draft version August 6, 2019

ABSTRACT The discrepancy between expected and observed cooling rates of X-ray emitting gas has led to the cooling flow problem at the cores of clusters of galaxies. A variety of models have been proposed to model the observed X-ray spectra and resolve the cooling flow problem, which involves heating the cold gas through different mechanisms. As a result, realistic models of X-ray spectra of galaxy clusters need to involve both heating and cooling mechanisms. In this paper, we argue that the heating time-scale is set by the magnetohydrodynamic (MHD) turbulent viscous heating for the Intracluster plasma, parametrised by the Shakura-Sunyaev viscosity parameter, α. Using a cooling+heating flow model, we show that a value of α ' 0.05 (with 10% scatter) provides improved fits to the X-ray spectra of cooling flow, while at the same time, predicting reasonable +0.63 cooling efficiency, cool = 0.33−0.15. Our inferred values for α based on X-ray spectra are also in line with direct measurements of turbulent pressure in simulations and observations of galaxy clusters. This simple picture unifies astrophysical accretion, as a balance of MHD turbulent heating and cooling, across more than 16 orders of magnitudes in scale, from neutron stars to galaxy clusters. Keywords: intracluster medium galaxies: clusters: individual (Hydra A, A2029, A2199, A496, A85) galaxies: cooling flow problem: active galactic nucleus (AGN):

1. INTRODUCTION inadequate and additional heating or cooling mechanisms The Intracluster Medium (ICM) of galaxy clusters consists should be incorporated into the model. Moreover, X-ray spec- of a plasma that is almost entirely ionized. This hot plasma ra- troscopy shows that the temperature drop toward the center is diates mostly in X-ray band which leads to a significant cool- limited to about a factor of three. The cooling seems to be ing of the ICM. At constant pressure, the cooling time of a frozen precisely in the region where we expect more rapid plasma is the gas enthalpy divided by the energy lost per unit cooling. In general, it appears that there is no strong evidence volume per unit time: for any significant amount of cold X-ray emitting gas (below 1/3 of the maximum temperature) in any cluster (Peterson & 5nkBT Fabian 2006). tcool ≡ , (1) 2nenH Λ(T,Z) There exist different manifestations of the cooling-flow problem: According to Peterson et al.(2003), there is the soft where Λ(T,Z) is the cooling function in terms of temperature X-ray cooling-flow problem and the mass sink cooling-flow T and metallicity Z, n is the particle number density, and kB is problem. The soft X-ray cooling-flow problem refers to the the Boltzmann’s constant. In the cores of clusters, the cooling discrepancy seen between the predicted and observed soft X- time dips below 5 × 108 yr, i.e. the inferred radiative cool- ray spectrum, e.g. the lack of expected emission lines from a ing time of the gas in the central part, where X-ray emission gas cooling to low temperatures at the core of the cluster. The is sharply peaked, is much shorter than the age of the clus- mass sink cooling-flow problem refers to the lack of colos- ter, which suggests the existence of cooling flow. The stan- sal mass deposition in cooling clusters from the hypothesized arXiv:1806.08822v2 [astro-ph.GA] 4 Aug 2019 dard cooling flow model can be derived by combining con- cooling-flow plasma. tinuity, Navier-Stokes and energy conservation equation that, Many mechanisms have been proposed to prevent the gas after simplification, leads to: from cooling to low temperatures at the centers of cooling flows, such as the electron thermal conduction (Zakamska & dL 5k 1dp X = M˙ B − . (2) Narayan 2003) , Mechanical heating of infalling gas in dense dT 2µmp ρdT core systems (Khosroshahi et al. 2004) and turbulent heat- In the case of constant pressure, we get the standard isobaric ing (Zhuravleva et al. 2014), though the lead suspect amongst cooling flow model: them is mechanical heating by Active Galactic Nuclei (AGN). AGN outbursts produce winds and intense radiation that can dL 5Mk˙ heat the gas. Produced weak shocks delay cooling of gas by X = B . (3) dT 2µm reducing gas density and increasing the total energy (David K p 2001; McNamara et al. 2005; Forman et al. 2005) or by com- X-ray spectroscopy has demonstrated that this model is pensating lost entropy of the gas (Fabian et al. 2005). More- over, viscous damping of sound waves generated by repeated * [email protected] AGN outbursts may represent a significant source of heating 2

(Fabian et al. 2003). Direct evidence for these sound waves !$ came from the spectra observed by the Hitomi X-ray satel- !" lite, which measured the plasma’s line-of-sight velocity dis- !%

!" persion of 164 ± 10 km/s within the core of Perseus cluster. !# This supports the hypothesis that turbulent dissipation of ki- !" netic energy can supply enough heat to offset gas from cooling !$ !' ! ! (Hitomi Collaboration et al. 2016). $ !$ # In this paper,we provide a simple yet accurate thermody- !$ !# namic model for cooling+heating (or CpH) flows, which cap- ! tures the balance between turbulent heating and cooling in " !& cluster cores (e.g., Zhuravleva et al. 2014). We then show !# ! that the model can simultaneously explain the X-ray spectra $ and the observed turbulent energy of the cluster cores, using a single parameter α ' 0.1, for the Shakura-Sunyaev viscosity parameter, while at the same time, predict reasonable cool- Figure 1. Schematic illustration of cooling flow in each annulus. Here, we ing efficiency. As such, this picture also unifies astrophysical model temperature distribution within each annulus as a result of in-situ cool- accretion across 16 orders of magnitude in scale, from kilo- ing and heating processes. metres (around neutron stars and stellar black holes) to kilo- parsecs (in cores of galaxy clusters). injection into the system by viscous heating. To estimate the heating time, we note that waves (and weak shocks) produced 2. DATA AND SPECTRAL EXTRACTION by AGcluster ofNs can travel at most by speed of sound. As For this study, we use a sample of galaxy clusters presented a result, sound crossing time is the shortest time scale in the by Hogan et al.(2017). The sample consists of 5 galaxy clus- ICM. We further assume that heating or viscous time should ters observed with Chandra X-ray Observatory over long ex- be a multiple of sound crossing time posure times. All five clusters have a central cooling time R r 3µm ≤ 1×109 yrs (Cavagnolo et al. 2009) suitable for our intended t α−3/2t R p , heat = sound = 3/2 = 3 (4) analysis. The data are obtained from the Chandra imaging α cs 5α kBT online repository and analyzed using CIAO version 4.7. Bad where R is the distance to the center of the cluster, and α < 1. pixels are masked out using the bad pixel map provided by the The α parameter quantifies the ratio of turbulent/magnetic to pipeline. Background flares are removed, and point sources thermal energy, and is very similar to the Shakura-Sunyaev are identified with the CIAO task WAVDETECT and masked viscosity parameter in accretion disks (Shakura & Sunyaev out in all subsequent analysis. Finally, the blank-sky back- 1973), as in a turbulent medium equipartition implies: grounds are extracted for each target, and the images are pre- 2 2 2 pared in the energy range 0.5–7.0 keV. In addition, cavities hv i ∼ hvAi ∼ αcs , (5) and filaments within ICM were masked clear, since these regions are usually out of equilibrium. where v and vA are turbulent and Alfven speeds. The heating time is then given by the ratio of thermal energy nkBT by Because the cooling instabilities usually occur at small 2 1/2 2 hv i (.10 kpc) radii, we desire finely binned spectra in the cen- turbulent heating rate ρhv i R : tral cluster regions. Our example clusters have deep Chandra 2 data; as a result, choosing of annuli for spectral extraction is nkBT Rc R t ∼ ∼ s ∼ α−3/2t . heat 2 3/2 2 3/2 3/2 = sound (6) limited by resolution rather than the number of counts. ρhv i /R hv i α cs For each example cluster, concentric circular annuli are centered at the cluster center. The width of the central an- The viscosity parameter typically takes a value of α ∼ 0.01 − nulus is 3 pixels, where each pixel is 0.492 arcsec across. The 0.1 in magnetohydrodynamic (MHD) simulations of weakly width of each annulus increases in turn by 1-pixel until the magnetized plasmas (e.g., Salvesen et al. 2016). sixth annulus, beyond which the width of each annulus is 1.5 Now, the central idea of our proposal is that the main driver times the width of the previous one with a total number of of thermal distribution in ICM is a balance of cooling and 10 annuli per source. For each OBSID we have spectra along- heating, governed by tcool/tsound (in contrast to, e.g., thermal side response matrix files (RMFs) and auxiliary response files instability determined by tcool/tfree−fall McCourt et al. 2012; 3/2 (ARFs). We keep spectra separate before fitting them, but at Hogan et al. 2017). Since tcool drops faster than T (at the time of running the XSPEC we load them simultaneously. constant pressure; see below) while theat ∝ tsound grows as Since emission from outer parts of the ICM affects and con- T −1/2 at low temperatures, the cold gas cools copiously, as taminates inter parts of spectra, we use deprojected spectra to in the standard cooling flows. However, the heating would obtain more accurate data. In order to fit observed data we win over the cooling for high temperatures. As a result, there load the extracted spectra for each cluster with their matched is thermodynamic equilibrium at the temperature T ∗ where ∗ ∗ response files into XSPEC version 12.9.1 (Arnaud 1996) and tcool(T ) ∼ theat(T ). In a (nearly) steady state (e.g., close to use fixed values of NH reported in (Main et al. 2015). the cluster core), most of the gas would sit near this temperature 2. However, gas with T T ∗ would cool down to form 3. MODIFIED COOLING+HEATING (CPH) FLOW MODEL: COOLING V.S. SOUND CROSSING atomic or molecular gas, and eventually stars. On the other hand gas with T T ∗ would heat up and eventually feed the AGNs outburst and jets can pump energy into the ICM (Mc- Namara & Nulsen 2007). This can be done by shock waves 2 Even though this is an unstable equilibrium, random turbulent motion or sound wave deposition close to the AGN. We introduce a can make it semi-stable, similar to the inverted pendulum with an oscillating tip (Kapitza’s pendulum; Landau & Lifshitz 1969). new timescale, theat, which represents time scale of the energy COOLING+HEATING FLOWSIN GALAXYCLUSTERS 3

Cluster NH z Scale Observation IDs Exposure (ks) M˙ SFR M˙ cool 22 −2 −1 −1 (10 cm ) (kpc/”) Cleaned M yr M yr 0.1 A2029 0.033 0.0773 1.464 891, 4977, 6101 103.31 0.80.09 269.2 ± 1.1 9 A2199 0.039 0.0302 0.605 10748, 10803, 10804, 10805 119.61 11 47.9 ± 1.1 0.01 A496 0.040 0.0329 0.656 931, 4976 62.75 0.180.01 56.2 ± 1.1 2.5 A85 0.039 0.0551 1.071 904, 15173, 15174, 16263, 16264 193.64 0.10.1 87.1 ± 1.0 7 Hydra A 0.043 0.0550 1.069 4969, 4970 163.79 42 109.6 ± 1.0

Table 1 Sample clusters from Chandra data (Hogan et al. 2017). Standard cosmology with H0= 70 km s−1 Mpc−1 has been used and scales are angular. The observed star formation rate ,M˙ SFR, and classical cooling rate, M˙ cool , is obtained from McDonald et al.(2018) and the K-band magnitude are collected from Gavazzi & Boselli(1996) and Jarrett et al.(2003). cosmic ray population, through Fermi acceleration. This in- situ mass loss within each annulus should be replenished by

4 Mkcflow a slow accretion/inflow of plasma from cluster outskirts, in 10 CpH steady state (see Figure1 for a visual representation). To see this more quantitatively, let’s begin with the cooling flow model. In the case of standard isobaric cooling flow model, we have:

5k n dT 10 5 B = −n n Λ(T,Z) (7) 2 dt e H or

d lnT 1 Emission Measure = − . (8) dt tcool 10 6 To modify this, we add a heating term to the equation (8) and rewrite it as: 1 10 20 d lnT 1 1 T(kev) = − + (9) dt tcool theat T The mass-weighted probability distribution of ln should Figure 2. The emission measure for seventh annulus of A2029 with using satisfy the conservation equation: mkcflow model and our CpH model. ∂P(lnT) ∂ d lnT + P(lnT) = 0, (10) 4. A NEW SPECTRAL MODEL ∂t ∂ lnT dt In the standard cooling flow model, the spectrum in steady which in steady state yields: state can be calculated using: ˙ d lnT 5MkB P(lnT) = A = const. (11) dLcool = nenH Λ(T,Z)dV = dT, (14) dt 2µmp Therefore the probability distribution takes the following where M˙ is the mass deposition rate, and µ is the mean molec- form: ular weight. We can re-express this equation in terms of a differential emission measure dEM = nenH dV which captures tcool the differential distribution of plasma across temperatures: PCpH (lnT) = A , (12) |1 − tcool | theat dEM 5Mk˙ = B . (15) where A is a normalization constant that is fixed by requiring dT 2µmpΛ(T,Z) total integrated probability 3 is 1: We can now convolve this distribution with the energy de- Z dκ −1 tcool A = d lnT . (13) pendent line power dE (E,T,Z) to produce an X-ray spec- tcool |1 − t | trum, which can be compared with observations. The spectral heat source model as a function of emission measure will be: Now, as discussed above, we see from Equation (12) that T ∗ d Z max dEM dκ when tcool theat (or T T ), we obtain the standard cooling ∗ = (E,T,Z)dT, (16) flow model. In contrast, if tcool theat (or T T ) we have dE 0 dT dE P(lnT) ≈ t . heat where Z ∞ dκ 3 We should note that, even though this integral is formally divergent at Λ(T,Z) = dE (E,T,Z)E. (17) tcool = theat, the divergence is only logarithmic, and is presumably regularized 0 dE by stochastic turbulent motion. For our calculation, the integral is regularized by the finite temperature bins in the spectral modeling. However, due to the Equation (16), which is the prediction of the standard cool- logarithmic nature of divergence, the choice of binning has little effect on our ing flow model, cannot provide a good fit to the X-ray spec- results. tra of galaxy cluster cores (or outskirts) (Peterson & Fabian 4

2006). In practice, it is common to use a single (or multi- after some simplification we can express pressure in terms of ∗ ) temperature model (so-called “mekal” in XSPEC) to fit the M˙ cool and T : X-ray spectra, even though this cannot be physically justified ˙ Z Tmax 3 3 given the short cooling times in cluster cores. 2 5M kBT d lnT p = f (T) (24) Our proposal to solve the problem is to apply the probabil- 2µmpV 0 Λ(T,Z)|1 − ∗ | ity distribution 12 introduced in the previous section, which f (T ) captures both heating and cooling in the flow. By plugging in We can further express ne and nH in terms of the total num- the explicit expressions for of tcool (Eq.1), theat (Eq.6) and ber density n, using n = ne + nH + nHe, X = nH /(nH + 4nHe) and pressure p = nkT we find: ne = nH + 2nHe, which yield

tcool 2X + 2 4X PCpH (lnT) = A f (T) , (18) ne = n, nH = n, (25) |1 − f (T ∗) | 5X + 3 5X + 3 where X is the hydrogen mass fraction and we shall consider where X ≈ 0.75. Now, by eliminating pressure p in the following s ∗ 5/2 2 2 α f T n T nenH 12µmp p R equation for (26) and ( ) (Eq. 19), using Eq. (25) for e f (T) ≡ , and f (T ∗) = (19) and n , we find α which gives the value of viscosity parame- Λ(T,Z) n2 α3k5 H 125 B ter, within the annulus at radius R and volume V, in terms of ∗ ˙ We can now modify the cooling flow emission measure the X-ray observables T and M. dEM 1 dEM → f (T) and apply it in the spectral model (16): 5 4 ∗ 2 −1/3 dT |1− ∗ | dT h 125k n f (T ) i f (T ) α = B (26) 12n2n2 µm p2R2 d Z Tmax 1 dEM dκ e H p = (E,T,Z)dT. (20) dE |1 − f (T) | dT dE We name our model Cooling plus Heating (CpH) and it is 0 f (T ∗) publicly available at: https://heasarc.gsfc.nasa. We fit the observed spectra after implementing our CpH gov/xanadu/xspec/manual/node158.html model into the XSPEC. To do this, we modify the emission 5. RESULTS measure of the standard cooling flow (or “mkcflow”) model. 4 To follow the changes in the emission measure after we im- In this section, we probe our CpH model using the Chan- plement our model into the XSPEC, we present an example dra clusters sample and demonstrate that they are superior of our best-fit emission measure of the cooling flow model, (or comparable) fits in cluster cores, in comparison to single- compared to our model, for the same cluster and the same an- temperature mekal models. We then briefly discuss the im- nulus in Figure2. In contrast to the mkcflow mode, which plications for the cooling efficiency and turbulent viscosity in has a smooth emission measure, our model has a clear peak in ICM. ∗ temperature (where T = T or theat = tcool), as well as extended We use XSPEC to fit Chandra X-ray data. For the sin- tails. gle temperature model (phabs×mekal) and our CpH model In order to fit observed Chandra spectra, T ∗ is treated as (phabs× CpH). We fix abundance to the solar and also fix a free parameter in XSPEC, while we fixed the lower and the the hydrogen column densities NH and redshifts to the values −2 provided in Table (1). We ran Markov Chain Monte Carlo upper limits of the integration Tmin = 10 keV and Tmax = 50 keV. As such, our spectral model has the same number of pa- (MCMC) to find best-fit parameters. The best-fit parameter for single temperature model is T and for our CpH model rameters as a single-temperature (or mekal) model. It is worth ∗ mentioning that if the lower limit value of mkcflow model is is T (indicating the peak of the CpH probability distribu- set to such a small value, it is impossible to fit the observed tion). The goodness of the fit and the best-fit parameters of data. In contrast, as we see below, we can find a good fit to our model, as well as the single temperature model, are provided in Table2. We notice that the peaks of the tempera- data using the modified emission measure (20). ∗ To obtain α we first compute the pressure , p within each ture distribution in our best-fit models, T , happen to be close annulus of volume V = 4/3π(r3 − r3 ), as follows: to the best-fit T in the single-temperature models. However, out in typically our model provides a better (or comparable) fit to dM data in cluster cores (with an acceptable χ2 for the number of P(lnT) = (21) data points). In the cluster outskirts, where the assumption of Mtot d lnT a steady state cooling/heating flow is not valid, none of the dM dV p dV models provide a good fit to the data. While the CpH and the = µmpn = µmp (22) mekal d lnT d lnT kBT d lnT models provide satisfactory fits to X-ray data for the same annuli, the CpH model is preferred at ∆χ2 = −56.9 (or By eliminating dM in Eq. (22) using Eq. (21)and integrating 7.5σ level) if we combine all the annuli with satisfactory fits. over volume we get the following relation for Mtot : We can now plug our best-fit CpH spectral model into Eq. (26) to find the MHD/turbulent viscosity parameter α, which µmp pV Mtot = R (23) is plotted in Figure3. We find that a value of α ' 0.05 (with a kBTP(lnT)d lnT small intrinsic scatter of ±10%) in the cluster cores (< 20-30 We notice that normalization of the Eq. (18), A, has the in- kpc, where CpH model can give a satisfactory fit to spectrum verse time dimension and as a result we define our mass de- in Table2). More precisely, measured α’s are consistent with position rate in each annulus as M˙ cool = AMtot . As a result a gaussian distribution with mean α¯ and scatter σ (see Figure 3b) 4 We modify the source codes of mkcflow model in XSPEC to match our desired emission measure. +0.005 +0.004 α¯ = 0.048−0.006, σ = 0.011−0.006, (27) COOLING+HEATING FLOWSIN GALAXYCLUSTERS 5

1.0 A85 A85 A496 A496 HydraA HydraA 1 A2199 10 A2199 A2029 A2029

100

0.05 Cold Fraction

10 1

0.01 10 2 1 10 100 1 10 100 R(kpc) R(kpc)

Figure 4. The predicted cold to hot gas density fraction accumulated over 7.7 Gyr (i.e. since z = 1) , assuming our best-ﬁt model and steady state. 0.030

star formation rate. The key parameter for mass deposition

0.025 rate is M˙ , which is directly provided by the spectral ﬁtting in the XSPEC. Another useful quantity to be calculated is the ratio of cold to hot gas density, or cold fraction, which

0.020 is 7.7 Gyr × A. We estimate the cold fraction over a period of ∼ 7.7 Gyr (i.e. since z = 1) for each cluster. The cold fraction is shown in Figure4, which indicates that most of the cold gas

0.015 is accumulated within the inner 10 kpc of cluster cores, where its density dominates the hot gas by up to an order of magnitude (we have used only “good ﬁts” from Table2, in Figs3

0.010 and4). Let us assume that the cooled gas is used as fuel for star for- M˙ SFR mation. We can define cooling efficiency as cool ≡ from M˙ cool 0.005 which we can specify how well AGN feedback can offset the runaway cooling. We have plotted M˙ SFR against M˙ cool for each hM˙ i . +0.8M / 0.000 cluster in Figure5, showing that cool CpH = 3 7−1.0 yr, 0.035 0.040 0.045 0.050 0.055 0.060 0.065 while hM˙ cooliclassical ≡ Mgas(r < rcool)/tcool = 114±0.5M /yr. Here in calculating M˙ cool for each cluster in the context of CpH model, we have considered and added the mass deposition rate of annuli that satisfy A−1 < 3Gyr. This condition Figure 3. (a) (top) The measured MHD/turbulent viscous heating parameter α, defined as the square of the ratio of sounds crossing to viscous dissipation ensures that the thermodynamic equilibrium in an annulus at 2/3 ∗ time α = (theat/tsound) . These values are inferred by fitting our CpH model temperature T has been established. On the other hand, in to the spectra and fluxes of deprojected X-ray data from Chandra clusters. (b) calculating the classical mass deposition rate M˙ cool, the cool- (bottom) The 68% and 95% confidence regions for the mean α¯ and gaussian ing radius r is defined as the radius within which cooling intrinsic scatter σ, assuming α = α¯ ± σ. cool time tcool < 3 Gyr (McDonald et al. 2018). The observed star where errors reflect 1σ uncertainties. formation rates and classical cooling rates are reported in Ta- ˙ +2.3 These values are consistent with viscosity parameters in ble1 in which the mean value of MSFR is 1.2−0.5. As a result shearing box accretion disk simulations Salvesen et al.(2016), we can calculate the cooling efficiency for underlying clusters +0.63 +0.02 as well as simulated (e.g., Gaspari & Churazov 2013) or ob- which leads to cool = 0.33−0.15 for CpH and cool = 0.01−0.004 served turbulent energy fraction in cluster cores (Zhuravleva using results of McDonald et al.(2018). Based on these re- et al. 2014, 2016; Hitomi Collaboration et al. 2016). sults we infer a much larger efficiency for star formation from CpH classical Considering that our proposed model is successful in rep- cooling in the CpH model, i.e. cool /cool ∼ 0.33/0.01 ∼ resenting the observed X-ray spectra, we can compare the 33 . mass deposition rate of our CpH model with the observed It worth mentioning that a clear advantage of our model 6

cosmology group meeting at Perimeter Institute for their useful comments. We also would like to thank Mike Hogan for providing us with the data used in Hogan et al.(2017) and CpH we thank the anonymous referee for his/her review and useful Classical cooling comments. The scientiﬁc results reported in this article are based on observations made by the Chandra X-ray Observa-

1 tory and has made use of software provided by the Chandra X- 10 ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research is supported in part by the Univer- )

1 sity of Waterloo, National Science and Engineering Council

r of Canada (NSERC), and Perimeter Institute for Theoretical y 100 Physics. Research at Perimeter Institute is supported by the M (

R Government of Canada through the Department of Innova- F S tion, Science and Economic Development Canada and by the M Province of Ontario through the Ministry of Research, Inno- vation and Science. 10 1 REFERENCES

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2 2 ∗ ˙ Cluster R(kpc) χCpH χmekal T (keV) Tmekal(keV) M(M /yr) Lx(erg/s) Net_counts Nd.o. f +0.4 +0.3 +0.04 +0.86 40 1.49±0.6 62 66 1.4−0.5 1.3 −0.4 0.1−0.05 6.76−0.88 × 10 3300 +1.1 +0.7 +0.06 +0.19 41 2.83±0.74 66 66 3.0−0.7 2.5−0.4 0.12−0.07 2.89−0.22 × 10 7203 +0.7 +0.5 +0.09 +0.34 41 4.46±0.89 60 61 2.8−0.5 2.4−0.3 0.26−0.12 5.35−0.27 × 10 11695 +0.4 +0.4 +0.16 +0.36 41 6.40±1.04 61 61 2.5−0.3 2.3−0.2 0.14−0.11 7.7−0.38 × 10 16671 +0.3 +0.2 +0.19 +0.4 41 A2199 8.63 ±1.19 76 82 2.9−0.5 2.5−0.2 0.75−0.28 16.5−0.5 × 10 21142 64 +0.3 +0.2 +0.28 +0.4 41 11.61 ±1.79 62 62 2.9−0.3 2.7−0.2 0.27−0.20 21.0−0.5 × 10 28883 +0.5 +0.3 +0.33 +0.65 41 16.07±2.68 63 66 2.8−0.6 2.5−0.3 0.73−0.41 18.7−0.52 × 10 38885 +0.6 +0.3 +0.33 +0.74 41 22.77±4.01 57 56 3.3−0.4 3.1−0.3 0.33−0.25 32.5−0.67 × 10 68880 +0.5 +0.3 +0.7 +0.89 41 32.97 ±6.18 47 48 4.4−0.5 3.9−0.3 1.1−0.7 76.5−0.93 × 10 127922 +0.2 +0.1 +1.1 +1.4 41 48.2±9.08 101 105 4.7−0.3 4.4−0.1 2.2−1.1 243−1.3 × 10 216253 +0.9 +0.5 +0.3 +0.35 41 5.00±1.3 19 28 3.7−0.7 3.1−0.4 1.6−0.4 9.5−0.33 × 10 7241 +1.0 +0.5 +0.4 +0.64 41 7.89±1.6 17 20 4−0.7 3.3−0.4 1.5−0.6 16.2−0.5 × 10 9862 +0.7 +0.4 +0.6 +0.5 41 11.31 ±1.8 21 22 3.0−0.6 2.5−0.4 1.5−0.9 16.0−0.6 × 10 16102 +2.0 +1.0 +0.4 +0.9 41 15.25±2.1 27 27 5−1.4 3.7−0.8 1.1−0.6 20.6−1.9 × 10 20226 +0.7 +0.4 +1.4 +1.0 41 HydraA 20.51±3.2 14 14 3.2−0.5 2.8−0.3 2.5−1.7 32.1−1.0 × 10 26833 21 +0.8 +0.3 +2.6 +1.2 41 28.40 ±4.7 22 23 3.5−0.3 3.1−0.2 3.7−2.6 73.1−1.1 × 10 44738 +0.6 +0.3 +3.3 +1.6 41 40.24±7.1 25 33 4.4−0.5 3.7−0.2 7.7−3.6 124−1.7 × 10 68412 +0.4 +0.1 +6.1 +1.9 41 58.25 ±10.9 36 42 4.5−0.1 4.2−0.1 8.2−3.9 258−2.1 × 10 111067 +0.3 +0.1 +6.7 +2.7 41 85.20±16.0 67 87 4.6−0.2 4.3−0.1 13.4−5.6 409−2.0 × 10 145225 +0.3 +0.1 +7.2 +2.0 41 125.2±23.9 45 64 4.6−0.2 4.2−0.1 14.4−7.2 388−1.9 × 10 153239 +0.3 +0.4 +0.24 +0.1 41 1.08±1.0 63 63 2.6−0.65 2.2−0.3 0.21−0.17 4.2−0.18 × 10 1993 +0.6 +0.4 +0.40 +2.0 41 3.60±1.44 103 103 3.0−0.4 2.8−0.3 0.31−0.24 35.2−3.6 × 10 12208 +0.5 +0.5 +0.67 +1.1 41 6.84±1.8 104 104 3.1−0.2 2.9−0.3 0.50−0.40 49.7−1.7 × 10 13500 +2.7 +3.2 +0.4 +2.8 41 10.80±2.16 105 106 11.7−3.4 7.8−1.8 1.3−0.6 91.4−3.0 × 10 22702 +2.2 +0.9 +1.6 +0.4 42 A2029 15.49 ±2.5 98 97 5.6−1.2 4.5−0.6 1.8−1.4 12.9−0.3 × 10 28949 86 +4.1 +1.2 +1.3 +0.3 42 20.89±2.88 110 110 9.6−3.0 6.4 −1.0 2.3−1.6 19.1−0.6 × 10 34104 +2.4 +0.8 +3.0 +0.5 42 28.09±4.32 77 77 7.2−1.3 6.0−0.5 2.9−2.2 33.3−0.5 × 10 56203 +1.7 +0.5 +3.7 +0.6 42 38.90 ±6.48 117 117 8.1−0.8 6.9−0.5 4.0−2.9 63.8−0.5 × 10 88105 +1.8 +0.5 +1.7 +0.8 42 55.10±9.72 101 101 7.2−1.9 7.4 −0.4 1.7−1.3 99.8−0.3 × 10 128336 +1.0 +0.4 +6.4 +1.0 42 79.77±14.95 104 102 8.9−0.8 7.8−0.3 3.6−3.5 165−1.0 × 10 171978 +0.6 +0.44 +0.06 +0.1 41 1.61±0.65 5 9 1.7−0.4 1.4−0.19 0.15−0.06 1.3−0.1 × 10 533 +0.2 +0.2 +0.10 +0.2 41 3.07 ±0.81 4 10 1.3−0.2 1.1−0.1 0.47−0.15 2.7−0.2 × 10 2156 +0.5 +0.2 +0.17 +0.4 40 4.84±0.97 6 8 1.8−0.2 1.7−0.2 0.50−0.22 5.6−0.4 × 10 4680 +1.2 +0.6 +0.20 +0.6 41 6.94±1.13 7 7 2.9−0.8 2.3−0.3 0.38−0.25 10.1−0.5 × 10 7361 +0.8 +0.5 +0.37 +0.5 41 A496 9.36±1.29 6 6 2.8−0.6 2.3−0.2 0.51−0.33 15.9−0.6 × 10 10129 10 +0.8 +0.2 +0.37 +0.8 41 12.59±1.94 12 11 2.4−0.2 2.2−0.1 0.52−0.39 32.6−0.7 × 10 13736 +0.3 +0.1 +0.26 +1.0 41 17.43±2.90 24 24 2.9−0.3 2.7 −0.1 0.37−0.29 48.6−0.8 × 10 21962 +0.4 +0.3 +0.55 +1.2 41 24.69±4.36 7 6 3.2−0.3 3.0−0.2 0.55−0.44 80.4−1.1 × 10 31017 +0.3 +0.2 +0.71 +1.2 41 35.74±6.70 24 23 4.1−0.5 3.8−0.2 0.71−0.57 122−1.3 × 10 46112 +0.6 +0.2 +1.1 +1.7 41 52.29±9.84 19 19 4.3−0.3 4.0−0.2 1.1−1.0 194−1.1 × 10 64190 +1.1 +0.7 +0.18 +0.4 41 2.63 ±1.1 23 22 2.4−0.6 1.98−0.3 0.34−0.17 5.2−0.4 × 10 3503 +1.1 +0.7 +0.29 +0.8 41 5.0±1.32 23 22 3−0.7 2.4−0.4 0.68−0.36 14.7−0.9 × 10 8458 +0.4 +0.4 +0.51 +1.2 41 7.90±1.58 38 39 2.7−0.4 2.2−0.2 1.69−0.79 32.3−1.4 × 10 11147 +0.4 +0.1 +0.90 +1.3 41 11.33±1.84 98 98 2.6−0.3 2.3−0.1 1.96−1.2 42.5−1.3 × 10 14413 +1.1 +0.6 +0.6 +1.2 41 A85 15.28±2.11 32 32 3.6−0.8 2.8−0.4 1.4−0.8 39.4−2.2 × 10 16205 24 +0.9 +0.6 +0.6 +2.2 41 20.55±3.16 22 28 3.3−0.6 2.8−0.4 2.3−0.6 52.0−1.5 × 10 26093 +1 +0.3 +0.7 +2.0 41 28.45±4.74 93 93 4.4−0.8 3.8−0.3 3.7−2.9 143−2.4 × 10 46348 +1.3 +0.2 +2.2 +3 41 40.31±7.11 73 73 5.8−1 4.6−0.2 3.3−2.2 223−2 × 10 68838 +1.2 +0.2 +3.2 +3.2 41 58.36 ±10.93 198 198 6.4−1 5.1−0.2 5.3−4.9 381−2.2 × 10 105746 +1.3 +0.2 +4.5 +4.7 41 85.36±16.10 278 278 6.7−0.7 5.8−0.2 3.4−2.2 556−3.5 × 10 134958

Table 2 Calculated best-fit χ2 of our CpH model and mekal single-temperature model, with the best-fit parameters of fitting for the first ten annuli. T ∗ and M˙ are best fit 2 parameters of our CpH model. The red (italic) fonts suggest that the fit is outside the 90% expected range for the (reduced) χ for Nd.o. f , i.e. it is not a good fit. We see that our CpH model typically provides a lower χ2, or a better fit, in cluster cores.

APPENDIX In the appendix we are providing temperature, density and ∆χ2 profiles of CpH and Mekal model. Figure6 shows the close proximity of the density profiles in CpH and Mekal Model (Data of Mekal profiles are extracted from Hogan et al.(2017)). As can be inferred from Figure6 the peak temperature profiles of CpH and Mekal temperatures profiles show almost the same trend. 2 2 2 In order to verify that the CpH model provides an improved fit over the Mekal model, we present the ∆χ = χCpH −χMekal profile in Figure7 in which we are using only good fits (black bins in Table2). 8

A85_mekal 14 A496_mekal HydraA_mekal A2199_mekal A2029_mekal 12 A85_CpH 10 1 A496_CpH HydraA_CpH A2199_CpH A2029_CpH

) 10 3 ) m c V 8 ( e y k t ( i s T

n 6 e 10 2

D A85_mekal A496_mekal HydraA_mekal 4 A2199_mekal A2029_mekal A85_CpH A496_CpH 2 HydraA_CpH A2199_CpH A2029_CpH 0 100 101 102 100 101 102 R(kpc) R(kpc)

Figure 6. Deprojected density and temperature proﬁles of CpH (solid lines) compared with what Hogan et al.(2017) found using Mekal (dotted lines).

4 2

8 A85 A496 10 HydraA A2199 A2029

101 102 R(kpc)

Figure 7. ∆χ2 proﬁle based on good ﬁts in Table2. It is apparent from this plot that the majority of bins have lower χ2 for CpH with respect to Mekal.